Gradient-based multi-objective optimization (MOO) is essential in modern machine learning, with applications in e.g., multi-task learning, federated learning, algorithmic fairness and reinforcement learning. In this work, we first reveal some limitations of Pareto stationarity, a widely accepted first-order condition for Pareto optimality, in the presence of sparse function-variable structures. Next, to account for such sparsity, we propose a novel solution concept termed Refined Pareto Stationarity (RPS), which we prove is always sandwiched between Pareto optimality and Pareto stationarity. We give an efficient partitioning algorithm to automatically mine the function-variable dependency and substantially trim non-optimal Pareto stationary solutions. Then, we show that gradient-based descent algorithms in MOO can be enhanced with our refined partitioning. In particular, we propose Multiple Gradient Descent Algorithm with Refined Partition (RP-MGDA) as an example method that converges to RPS, while still enjoying a similar per-step complexity and convergence rate. Lastly, we validate our approach through experiments on both synthetic examples and realistic application scenarios where distinct function-variable dependency structures appear. Our results highlight the importance of exploiting function-variable structure in gradient-based MOO, and provide a seamless enhancement to existing approaches.